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Difference between revisions of "2018 USAJMO Problems/Problem 5"

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==Problem 5==
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Let <math>p</math> be a prime, and let <math>a_1, \dots, a_p</math> be integers. Show that there exists an integer <math>k</math> such that the numbers <cmath>a_1 + k, a_2 + 2k, \dots, a_p + pk</cmath>produce at least <math>\tfrac{1}{2} p</math> distinct remainders upon division by <math>p</math>.
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==Solution==

Revision as of 01:03, 21 April 2018

Problem 5

Let $p$ be a prime, and let $a_1, \dots, a_p$ be integers. Show that there exists an integer $k$ such that the numbers \[a_1 + k, a_2 + 2k, \dots, a_p + pk\]produce at least $\tfrac{1}{2} p$ distinct remainders upon division by $p$.


Solution

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